I am afraid you will have to transfer the files yourself.
The best way to minimise downtime is to create a free dreamhosters.com sub-domain and set it to mirror your domain. You can then transfer your site files and ensure that everything works ok before disabling the old site.
Once everything is working OK at DreamHost, you can configure the domains DNS records to point at the DreamHost DNS servers (ns1.dreamhost.com etc...), you do this using whatever configuration utility your current domain registrar provides. Within a couple of days the DNS changes will propagate across the Internet and your new site will replace the old one.
Once all this is complete, you can transfer the domain registration to DreamHost as well, if this is what you want.
There is a wiki article on this procedure at the link below.
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